reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem Th38:
  for T being UnContinuous TopGroup, a being Element of T, W being
  a_neighborhood of a" ex A being open a_neighborhood of a st A" c= W
proof
  let T be UnContinuous TopGroup, a be Element of T, W be a_neighborhood of a";
  reconsider f = inverse_op T as Function of T, T;
  f.a = a" & f is continuous by Def7,GROUP_1:def 6;
  then consider H being a_neighborhood of a such that
A1: f.:H c= W by BORSUK_1:def 1;
  a in Int Int H by CONNSP_2:def 1;
  then reconsider A = Int H as open a_neighborhood of a by CONNSP_2:def 1;
  take A;
  let x be object;
  assume x in A";
  then consider g being Element of T such that
A2: x = g" and
A3: g in A;
  Int H c= H & f.g = g" by GROUP_1:def 6,TOPS_1:16;
  then g" in f.:H by A3,FUNCT_2:35;
  hence thesis by A1,A2;
end;
